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Unverified Commit 84cf7c6e authored by rrueger's avatar rrueger
Browse files

Algebra I: Commutative Rings. Rephrase Universal property of rings

parent 42d5428d
......@@ -332,17 +332,18 @@
\textbf{isomorphism}.
\end{definition}
\begin{theorem}
Let $R$ and $S$ be commutative rings, and let $\varphi: R \to S$ be a
homomorphism. If $s_1, \ldots, s_n \in S$, then there exists a unique
\begin{theorem}[Universal property of Rings]
Let $R$ and $S$ be commutative rings, and let $\varphi \colon R \to S$ be a
homomorphism. For fixed $s_1, \ldots, s_n \in S$, there exists a unique
homomorphism
%
\begin{align*}
\Phi: R[x_1, \ldots, x_n] \to S
\Phi \colon R[x_1, \ldots, x_n] & \to S \\
x_i & \mapsto s_i
\end{align*}
%
with $\Phi(x_i) = s_i$ for all $i$ and $\Phi(r) = \varphi(r)$ for all $r \in
R$.
with $\Phi(r) = \varphi(r)$ for all $r \in R$. That is, $\Phi$ uniquely
extends $\varphi$ from $R$ to $R[x_1, \ldots, x_n]$, sending $x_i$ to $s_i$.
\end{theorem}
\begin{definition}[Evaluation map]
......
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